Group theory is only interested in how the operations combine. It doesn't care what the set and the operations stand for. The main exception are permutation groups. They are of particular interest because every group
is isomorphic to a permutation group.

It turns out to be extremely difficult to map out the world of groups, even in the finite case. Group theory distinguishes all kinds of relations between groups (one group being a subgroup of another, one group being a normal divisor of another, etc.), properties related to group-to-group mappings (group homomorphisms), and so forth. Specific classes of groups have their own theories, with specific concepts and theorems.

Group theory is extremely abstract and often mentioned as a paradigmatic example of pure mathematics. The discovery of a practical application for Lie groups, one of those specific classes developed in theory, is often cited in defense of the need for society to support pure science.

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A non-scary explanation of group theory for non-mathematicians

This is an attempt at a simplistic explanation of group theory for non-mathematicians. If you looked at the above write-ups and got scared by all the long words and symbols, then this is for you. This is my first attempt at something like this, so I'd appreciate some guidance - if you get confused or bored reading this then please /msg me explaining the problem. Anyway, on with the show...

The example

Normally, mathematicians start by describing things in the most abstract way possible (see TMA's write-up under group for a relevent example), but that can be confusing until you're used to it. So here I'll present the theory through an example. The example I'm going to use is of the Rubik's cube, so if you don't know what one is please read that node, and if you've got one go and get it. OK? Right. Now a group is a bunch of things, just as in "a group of people", so I'd better tell you what things this group is made up of. Now play with the cube a bit, either really if you've got one or else mentally (don't worry if you're cubeless - I'm not going to be asking you to do anything complicated you can't do in your head).

So let's say your cube is in one particular state (which for simplicity we'll assume is the nice, solved state with each face one colour, not that it really matters) and imagine turning the faces round a bit. So although it's still a cube, the faces now have different colours on them. Now we're going to call what you've just done a "move", and the group we're looking at is just all the possible moves you could have made. OK?

Now, we'll say two moves are the same if they mean you end up at the same state. So turning the nearest face round 90oclockwise, and then doing it again, is the same thing as just turning it round once 180o either clockwise or anti-clockwise, and they are also the same as turning it round twice by 90o anti-clockwise. You end up with the same thing whichever of the three you do.

OK, so we've defined the things we're talking about. But a group in the mathematical sense isn't just a bunch of things, it's a bunch of things with a few special properties - so let's show that this group has them.

The four properties of a group

Closure

Firstly, we have to have some way of combining moves. Here, the obvious meaning for the combination of moves is to do one and then the other, and that's the meaning we'll use. Note it is important which we do first - try turning one face and then another one next to it, look at what you get and then put them back, and then try turning the same faces but in the opposite order. You'll get something quite different. Just in case you happen to like jargon, this means the group is "non-commutative", but if you don't please ignore this sentence. Hurry on, nothing to see here...

Now if you combine two moves - that is if you do one and then the other - then that combination is clearly a move in itself. This first property is called "closure", and just means that a combination of things in a group is still in the group.

Identity

Secondly, consider one special "move", which is to do absolutely nothing. Now you might think it a bit strange to call "nothing" a thing, never mind a move, but then simply by naming it "nothing" we've basically decided it's a thing anyway. OK, so that's a move in our group, and note that it is the same move as turning a face around 360o in either direction, or indeed as turning it round 180o twice. Now, this here nothing move is a special move, which has the following property - if you do something, and then nothing, then that's the same as just doing the something, which is also the same as doing nothing and then that something.

Which is of course just something very obvious put in a stupid way, but it's important and for a group to be a group it must have a special thing like our "nothing" move, what's called the identity. But it's not always something we could call "nothing" - for example, numbers form a group where the combination of two numbers is them multiplied together, and here the identity is 1. For example, 1 X 5 = 5 = 5 X 1.

Inverse

But away from nasty numbers and back to the cube, and the third thing we need for the moves to be a group. It's this simple - if we make a move, then we can undo it. However long and involved the move was, however many faces you had to turn, you can get back to where you started just by turning those same faces in the opposite direction and in the opposite order. So for any move, we have another move such that if we do one and then the other then it's just like not doing anything at all - in fact, such that the combination of those two moves, in either order, is the same as our "nothing" move. This second "backwards" move is called the "inverse". The equivalent idea for multiplication is where you take "one over" a number - 1/2 X 2 = 2 X 1/2 = 1, so 1/2 is the inverse of 2.

Associativity
One last thing, and we're done. Now this is something that's obvious for our case of cube moves, but in general is quite a weird thing, and it's called "associativity". With the cube all it is is this - if we've got three moves, and if we do the third move to a cube to which we've just done the first and then the second, then that's the same as doing the second and then the third moves to a cube on which we've just done the first.

Think about this a bit - it's actually quite an obvious thing once you understand what it means, but it's damned hard to describe. The basic point is that it doesn't matter how you group the moves when you talk about them, what matters is just which moves you make and in which order. The equivalent thing for multiplication of numbers is that, for example, (2 X 3) X 4 = 2 X (3 X 4) (i.e. 6 X 4 = 2 X 12). Which means we can just write "2 X 3 X 4" to mean either of those, just like we can say we're doing "the first then the second then the third" moves to mean either of the two things in the last paragraph.

Wahey!

So that's it. We've got a bunch of things - the moves on a Rubik's cube with combination defined as doing one then the other - such that they are closed, there is an identity move, each move has an inverse, and moves are associative. This means we can officially call the moves a "group". Well isn't that just great, I hear you say, but why exactly should we care? Well, for far too many reasons for me to be able to explain here, is the basic answer - though if you look around the various group theory nodes here on E2 you'll see there's a whole lot of stuff, albeit largely incomprehensible to non-mathematicians, which you can say about groups - all of which comes from those four properties we've just discussed.

A taster

I'll give you one little cool thing we can prove about groups here now, and if this gets a decent response maybe I'll write some more. What I'm going to show now, in similarly hopefully comprehensible language, is the answer to the following question. What happens if you keep making the same move on the cube? In particular, do you ever just end up back where you started? Think about this, I don't think the answer's immediately obvious.

Well, the answer is that, yes, we always end up back where we started if we keep making the same move for long enough. We can also say something significant about how many moves it will take before we get there, though that's the subject of another node. I'll prove the main thing though - be sure you understand what I've said already about the four properties of a group before reading the following. Also, I'm afraid I've no alternative but to use the odd symbol in this proof, so be prepared. But don't worry, although it might look a bit like it this still isn't real real maths.

The proof

First off, note that this is a "finite" group we're talking about, that is, there are only so many moves. Don't worry exactly how many there are (43,252,003,274,489,856,000, if you must know), the important thing is just that there is a specific number of moves, however huge that number may be. One easy way to see this is to note that the way we've set up what we're calling a move, and in particular what makes two moves the same, means that each move is equivalent to a particular colouring of the cube. There is only one distinct move that could get you to one way the little coloured stickers are arranged on the cube, since that's precisely how we defined what makes two moves the same. And it should be pretty obvious that there are only so many ways of positioning the stickers, so we've got a finite number of moves. By the way, in case you doubt that we could ever have anything else, try to decide how many numbers there are in the group of our multiplication example.

OK, so there are only so many moves, so if we keep on doing one particular move it can't keep on giving us new things forever. So if you think about it a bit, you'll see that what we get must either look like this :

START-->--W-->--...-->--START-->--W-->--...-->--START-->--W-->--...

where the lines with an arrow represent doing our move, the things in between are the states of the cube after that lot of moves, and the ... means we just keep on doing the move a load of times. So here we have a big long loop which always comes back to where it started and then repeats. It either looks like that or like this :

That is, again it loops but this time it doesn't loop right back to the start, but just back to some previous state and it keeps on looping from there.

Now, if we can show that the first case always happens and never the second, then we're sorted and we always get back to the start if we keep on making the same move. So we need to show the second case never happens. Well, suppose it did, so as in the picture above we've got some state Y from which it starts looping - that is our move moves some state in the chain after Y into Y. Here we've called that state V. So we must also have another, different state which our move moves into Y - the one just before the first Y that appears in the chain. Here we've called it X.

So we have that our one move moves both V and X, which are different, into one state Y. Now this should seem pretty messed up, and it is. Remember that we said each move had an inverse, another move which undoes it? So what happens if we apply that inverse move to Y? Well, on the one hand it's got to take us back to V, and on the other to X - but they're different, and clearly one move can't take us to two different things, so that can't happen. So the second case, where it loops but not from the start, can never happen. So it always loops from the start, and so always if you keep on making one move you'll get back to where you started, which completes the proof.

Conclusion

Well, huzzah. If you managed to understand all that, well done. If, what's more, you found the continual reference to the cube annoying when clearly the proof used far more abstract notions which apply more generally, then maybe you should consider taking up Maths. You'd be right, by the way - this does apply to all groups, and the proof is essentially the same (minus the references to "states", which I just put in because I think it makes things more concrete and easier to understand). But this one example is also pretty useful - it means that if you've got a pristine cube you can start messing it up and as long as you keep on doing the same thing, you can be sure it'll end up ok again (eventually, possibly after hundreds of moves though). You can also wow your more easily impressed friends with this, since it does seem kinda magical when the cube recovers from a long period of messed-upedness, and makes the cubist almost look like they know what they're doing, and not for example be just applying the simplest of group theoretical knowledge.

Addendum - a slightly more scary explanation of group theory for wannabe mathematicians

Although if you've understood this so far you should have grasped the fundamentals of group theory, there's still one major thing blocking you from understanding further. And that thing is... notation. Now this is where this writeup turns from non-scary to scary, which is why I've left it 'till the end, but hang on in there and you should be fine. All I'm going to do is present the shorthand that mathematicians use when talking about groups.

Firstly, I said that a group is a bunch of things. Now in maths, we call a bunch of things a "set", and one of the things of which it is a bunch an "element" of the set. We tend to denote a set by a capital letter and an element by a small one, so we might for example talk about an element g of a group G. In the example above g might be the move consisting of rotation of the red face by 90o clockwise, for example.

Now, we have said that in a group we can combine two of its elements to get another element. We write the combination of two elements g and h (in that order) as g*h or, more often, just as gh. So the property I've just mentioned would be written as
For all g and h in G, gh is in G

Remember the second property - the existence of an identity. We tend to call the identity element of a group e, or occasionally 1, or even 0. I'll use e, which is most common. So we have that
For all g in G, ge = eg = g

Now we come to the inverse. We write the inverse of g as g-1, so the property is this :
For all g in G there exists a g-1 in G such that gg-1 = g-1g = e

And the fourth property, associativity, is written like this -
For all a, b, c in G, (ab)c = a(bc)

Check you understand how that last line fits in with how I presented associativity for the cube above. If you can understand that, then you're pretty much sorted! One last thing - we write g2 for gg, g3 for ggg and so on. So the theorem we just proved can be presented like this -
If G is a finite group, and if g is an element of G, then there exists a positiveintegern such that gn = e(a positive integer is just an ordinary counting number, like 1 or 5 or 10998521)

Well, if you've understood all this then you're well on the way to "getting" group theory. If you do want to learn more, I recommend you read the following nodes roughly in the following order. If you see any terms you don't understand, hopefully they'll be hard-linked to something you can understand. And if they aren't, I recommend you /msg the author to get them so. Particularly if that author is me. Also you might want to read Set theory notation for some more basic notation these w/u's might use. Anyway, here's my list -

The other write-ups on this node, which will hopefully be a bit more comprehensible at this point

group, which has a concise presentation of the group axioms (the ones which took me a couple of pages writing in non-mathsy language)

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Group theory is often introduced a bit imprecisely. Perhaps this is because its practitioners are too familiar and comfortable with it. I will introduce a concise definition of a group and will prove a few important consequences of the definition.

A group is a set of elements, G, with a mapping G x G --> G (i.e. all pairs of elements in G are mapped to an element of G). The mapping of an element pair (a,b) to an element c is indicated by the expression ab = c. The mapping, called the group product, must obey the following rules:

One could imagine that a represents 180-degree rotation about the x-axis, b represents 180-degree rotation about the y-axis, c represents 180-degree rotation about the z-axis, e represents no rotation, and the group product means that one operation is followed by another. You can perform these rotations with a book in your hand to visualize them. Rotations about different axes are in general not commutative. You can verify this by performing 90-degree rotations with the book.